Since I'm a technical person and I'm now in a different part of our organization I had to do some digging around to research the current state of Q and O to formulate the following answer for you... Please don't take this as an authoritative or all encompassing response, but at a high-level...
I generally think of Vertex's product-lines as suites - in which users can select those components that make the most business sense.
Q Series is our client-server solution. We offer PL/SQL or C API integration interfaces. The user interface is a desktop client. Supports North American transaction (Sales & Use taxes and Canadian GST, PST)
O Series is our web-based solution. We offer a webservice API and a JAVA API for integration. The user interface is a web browser based client. Supports North American transaction (Sales & Use taxes and Canadian GST, PST as well as VAT). O-Series supports Accrual, AP Invoice, AR Billing, Asset Movement, Buyer Input Tax, Distribute Tax, ERS, Inventory Removal, Invoice, Invoice Verification, Purchase Order, Quotation, and Reversal transaction types. O Series can be installed on premise or accessed via the cloud.
Our JDE integration works with specific modules within JDE (this is true for both Q and O Series) - the specific modules each supports would need to be explained by our sales team.
Both offer powerful customization tools to configure rules for products and customers - the O Series product has an extremely powerful tax assist module that allows users to create sophisticated pre-processing and/or post-processing of transactions.
Address cleansing is available for both solutions, and generally it is a good idea to use this so the jurisdiction identification component of O Series and Q Series can best resolve where taxes should be calculated.
Both record transactions that are processed in a journal/register from which transactions can be processed by our North American Returns product. O Series is also designed to integrate with our Global International Tax Returns product.
Since others will undoubtedly discover this thread later here's some information that goes beyond the JDE integration:
For both we provide a large number of ERP and eCommerce integrations (again contact Vertex Sales for specific vendors and versions).
We have an O Series cloud offering that includes both tax calculation as well as tax returns for businesses that want a streamlined end-to-end transaction (indirect) tax package.
Overall O Series has a wider range of features... but Q Series is very capable for businesses that have needs that fit its features.
In any case - please consider giving our sales department a call so that they can understand your requirements and help you determine which combination or products makes the best sense for your business...
Thanks